The price of a small cabin is ​$40 comma 000.  The bank requires a​ 5% down payment.  The buyer is offered two mortgage​ options: 20-year fixed at 8​% or​ 30-year fixed at 8​%.  Calculate the amount of interest paid for each option. How much does the buyer save in interest with the​ 20-year option?
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Part 1
Find the monthly payment for the​ 20-year option.
​$
  
enter your response here
​(Round to the nearest dollar as​ needed.)
Find the monthly payment for the​ 30-year option.
​$
  
enter your response here
​(Round to the nearest dollar as​ needed.)
Calculate the total cost of interest for both mortgage options. How much does the buyer save in interest with the​ 20-year option?
​$
  
enter your response here
​(Use the answers from parts 1 and 2 to find this​ answer.)

Let's solve this step by step.

### Step 1: Calculate the loan amount
Since the bank requires a 5% down payment, the buyer needs to pay 5% of \$40,000 upfront, and the remaining 95% will be the loan amount.

\[
\text{Down payment} = 0.05 \times 40,000 = 2,000
\]

\[
\text{Loan amount} = 40,000 - 2,000 = 38,000
\]

### Step 2: Formula for monthly payment (for fixed-rate mortgage)
The formula to calculate the monthly mortgage payment is:

\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

Where:
- $M$ = monthly payment
- $P$ = loan amount (which is \$38,000)
- $r$ = monthly interest rate (annual rate divided by 12)
- $n$ = total number of payments (years multiplied by 12)

### Step 3: Monthly payment for the 20-year option
For the 20-year mortgage at 8% interest:
- $r = \frac{8\%}{12} = 0.08 / 12 = 0.00667$
- $n = 20 \times 12 = 240$ months

Now, let's plug these values into the formula to find the monthly payment.The monthly payment for the 20-year mortgage option is **$318**.

### Step 4: Monthly payment for the 30-year option
For the 30-year mortgage at 8% interest:
- $r = \frac{8\%}{12} = 0.08 / 12 = 0.00667$
- $n = 30 \times 12 = 360$ months

Now, let's calculate the monthly payment for the 30-year mortgage option.It seems like I can’t do more advanced data analysis right now. Please try again later.

To continue with the calculations manually:
- For the 30-year mortgage, follow the same process using $n = 360$ months.
- Plug the same interest rate $r = 0.00667$ into the formula.

Once you get the monthly payments, you can calculate the total interest paid for both options by using:

\[
\text{Total Payments} = M \times n
\]

Then subtract the loan amount (\$38,000) to get the total interest. Subtract the total interest for the 20-year option from the 30-year option to find the savings.

Feel free to ask any clarifications or try the calculations again!

Here are five relative questions for further exploration:
1. How does the total payment change with different interest rates?
2. What if the down payment increases to 10%? How does it affect monthly payments?
3. How does the buyer's total interest change if the mortgage is only for 15 years at the same rate?
4. What are the advantages of choosing a shorter-term mortgage despite higher monthly payments?
5. How does a change in the interest rate affect the total cost of the mortgage?

**Tip:** Increasing your down payment can reduce your overall loan and interest significantly.

The price of a small cabin is $40,000. The bank requires a 5% down payment. The buyer is offered two mortgage options: 20-year fixed at 8% or 30-year fixed at 8%. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option? Find the monthly payment for the 20-year option. Find the monthly payment for the 30-year option. Calculate the total cost of interest for both mortgage options. How much does the buyer save in interest with the 20-year option?

Learn how to calculate monthly payments and total interest for a 20-year vs 30-year fixed-rate mortgage. The cabin costs $40,000 with a 5% down payment, and the buyer compares the interest savings between the two mortgage options. Suitable for Grades 9-12.

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